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## G = D4⋊Dic14order 448 = 26·7

### 2nd semidirect product of D4 and Dic14 acting via Dic14/Dic7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4⋊Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4×Dic7 — D4×Dic7 — D4⋊Dic14
 Lower central C7 — C14 — C2×C28 — D4⋊Dic14
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4⋊Dic14
G = < a,b,c,d | a4=b2=c28=1, d2=c14, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 532 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D42Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C22×Dic7, D4×C14, C4.Dic14, Dic7⋊C8, C8⋊Dic7, D4⋊Dic7, C7×D4⋊C4, C28⋊Q8, D4×Dic7, D4⋊Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8⋊C22, Dic14, C22×D7, D42Q8, C2×Dic14, D4×D7, D42D7, C22⋊Dic14, D8⋊D7, D7×SD16, D4⋊Dic14

Smallest permutation representation of D4⋊Dic14
On 224 points
Generators in S224
(1 90 49 219)(2 220 50 91)(3 92 51 221)(4 222 52 93)(5 94 53 223)(6 224 54 95)(7 96 55 197)(8 198 56 97)(9 98 29 199)(10 200 30 99)(11 100 31 201)(12 202 32 101)(13 102 33 203)(14 204 34 103)(15 104 35 205)(16 206 36 105)(17 106 37 207)(18 208 38 107)(19 108 39 209)(20 210 40 109)(21 110 41 211)(22 212 42 111)(23 112 43 213)(24 214 44 85)(25 86 45 215)(26 216 46 87)(27 88 47 217)(28 218 48 89)(57 149 122 192)(58 193 123 150)(59 151 124 194)(60 195 125 152)(61 153 126 196)(62 169 127 154)(63 155 128 170)(64 171 129 156)(65 157 130 172)(66 173 131 158)(67 159 132 174)(68 175 133 160)(69 161 134 176)(70 177 135 162)(71 163 136 178)(72 179 137 164)(73 165 138 180)(74 181 139 166)(75 167 140 182)(76 183 113 168)(77 141 114 184)(78 185 115 142)(79 143 116 186)(80 187 117 144)(81 145 118 188)(82 189 119 146)(83 147 120 190)(84 191 121 148)
(1 104)(2 36)(3 106)(4 38)(5 108)(6 40)(7 110)(8 42)(9 112)(10 44)(11 86)(12 46)(13 88)(14 48)(15 90)(16 50)(17 92)(18 52)(19 94)(20 54)(21 96)(22 56)(23 98)(24 30)(25 100)(26 32)(27 102)(28 34)(29 213)(31 215)(33 217)(35 219)(37 221)(39 223)(41 197)(43 199)(45 201)(47 203)(49 205)(51 207)(53 209)(55 211)(57 71)(58 164)(59 73)(60 166)(61 75)(62 168)(63 77)(64 142)(65 79)(66 144)(67 81)(68 146)(69 83)(70 148)(72 150)(74 152)(76 154)(78 156)(80 158)(82 160)(84 162)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)(113 169)(114 128)(115 171)(116 130)(117 173)(118 132)(119 175)(120 134)(121 177)(122 136)(123 179)(124 138)(125 181)(126 140)(127 183)(129 185)(131 187)(133 189)(135 191)(137 193)(139 195)(141 170)(143 172)(145 174)(147 176)(149 178)(151 180)(153 182)(155 184)(157 186)(159 188)(161 190)(163 192)(165 194)(167 196)(198 212)(200 214)(202 216)(204 218)(206 220)(208 222)(210 224)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 115 15 129)(2 114 16 128)(3 113 17 127)(4 140 18 126)(5 139 19 125)(6 138 20 124)(7 137 21 123)(8 136 22 122)(9 135 23 121)(10 134 24 120)(11 133 25 119)(12 132 26 118)(13 131 27 117)(14 130 28 116)(29 70 43 84)(30 69 44 83)(31 68 45 82)(32 67 46 81)(33 66 47 80)(34 65 48 79)(35 64 49 78)(36 63 50 77)(37 62 51 76)(38 61 52 75)(39 60 53 74)(40 59 54 73)(41 58 55 72)(42 57 56 71)(85 147 99 161)(86 146 100 160)(87 145 101 159)(88 144 102 158)(89 143 103 157)(90 142 104 156)(91 141 105 155)(92 168 106 154)(93 167 107 153)(94 166 108 152)(95 165 109 151)(96 164 110 150)(97 163 111 149)(98 162 112 148)(169 221 183 207)(170 220 184 206)(171 219 185 205)(172 218 186 204)(173 217 187 203)(174 216 188 202)(175 215 189 201)(176 214 190 200)(177 213 191 199)(178 212 192 198)(179 211 193 197)(180 210 194 224)(181 209 195 223)(182 208 196 222)

G:=sub<Sym(224)| (1,90,49,219)(2,220,50,91)(3,92,51,221)(4,222,52,93)(5,94,53,223)(6,224,54,95)(7,96,55,197)(8,198,56,97)(9,98,29,199)(10,200,30,99)(11,100,31,201)(12,202,32,101)(13,102,33,203)(14,204,34,103)(15,104,35,205)(16,206,36,105)(17,106,37,207)(18,208,38,107)(19,108,39,209)(20,210,40,109)(21,110,41,211)(22,212,42,111)(23,112,43,213)(24,214,44,85)(25,86,45,215)(26,216,46,87)(27,88,47,217)(28,218,48,89)(57,149,122,192)(58,193,123,150)(59,151,124,194)(60,195,125,152)(61,153,126,196)(62,169,127,154)(63,155,128,170)(64,171,129,156)(65,157,130,172)(66,173,131,158)(67,159,132,174)(68,175,133,160)(69,161,134,176)(70,177,135,162)(71,163,136,178)(72,179,137,164)(73,165,138,180)(74,181,139,166)(75,167,140,182)(76,183,113,168)(77,141,114,184)(78,185,115,142)(79,143,116,186)(80,187,117,144)(81,145,118,188)(82,189,119,146)(83,147,120,190)(84,191,121,148), (1,104)(2,36)(3,106)(4,38)(5,108)(6,40)(7,110)(8,42)(9,112)(10,44)(11,86)(12,46)(13,88)(14,48)(15,90)(16,50)(17,92)(18,52)(19,94)(20,54)(21,96)(22,56)(23,98)(24,30)(25,100)(26,32)(27,102)(28,34)(29,213)(31,215)(33,217)(35,219)(37,221)(39,223)(41,197)(43,199)(45,201)(47,203)(49,205)(51,207)(53,209)(55,211)(57,71)(58,164)(59,73)(60,166)(61,75)(62,168)(63,77)(64,142)(65,79)(66,144)(67,81)(68,146)(69,83)(70,148)(72,150)(74,152)(76,154)(78,156)(80,158)(82,160)(84,162)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111)(113,169)(114,128)(115,171)(116,130)(117,173)(118,132)(119,175)(120,134)(121,177)(122,136)(123,179)(124,138)(125,181)(126,140)(127,183)(129,185)(131,187)(133,189)(135,191)(137,193)(139,195)(141,170)(143,172)(145,174)(147,176)(149,178)(151,180)(153,182)(155,184)(157,186)(159,188)(161,190)(163,192)(165,194)(167,196)(198,212)(200,214)(202,216)(204,218)(206,220)(208,222)(210,224), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,115,15,129)(2,114,16,128)(3,113,17,127)(4,140,18,126)(5,139,19,125)(6,138,20,124)(7,137,21,123)(8,136,22,122)(9,135,23,121)(10,134,24,120)(11,133,25,119)(12,132,26,118)(13,131,27,117)(14,130,28,116)(29,70,43,84)(30,69,44,83)(31,68,45,82)(32,67,46,81)(33,66,47,80)(34,65,48,79)(35,64,49,78)(36,63,50,77)(37,62,51,76)(38,61,52,75)(39,60,53,74)(40,59,54,73)(41,58,55,72)(42,57,56,71)(85,147,99,161)(86,146,100,160)(87,145,101,159)(88,144,102,158)(89,143,103,157)(90,142,104,156)(91,141,105,155)(92,168,106,154)(93,167,107,153)(94,166,108,152)(95,165,109,151)(96,164,110,150)(97,163,111,149)(98,162,112,148)(169,221,183,207)(170,220,184,206)(171,219,185,205)(172,218,186,204)(173,217,187,203)(174,216,188,202)(175,215,189,201)(176,214,190,200)(177,213,191,199)(178,212,192,198)(179,211,193,197)(180,210,194,224)(181,209,195,223)(182,208,196,222)>;

G:=Group( (1,90,49,219)(2,220,50,91)(3,92,51,221)(4,222,52,93)(5,94,53,223)(6,224,54,95)(7,96,55,197)(8,198,56,97)(9,98,29,199)(10,200,30,99)(11,100,31,201)(12,202,32,101)(13,102,33,203)(14,204,34,103)(15,104,35,205)(16,206,36,105)(17,106,37,207)(18,208,38,107)(19,108,39,209)(20,210,40,109)(21,110,41,211)(22,212,42,111)(23,112,43,213)(24,214,44,85)(25,86,45,215)(26,216,46,87)(27,88,47,217)(28,218,48,89)(57,149,122,192)(58,193,123,150)(59,151,124,194)(60,195,125,152)(61,153,126,196)(62,169,127,154)(63,155,128,170)(64,171,129,156)(65,157,130,172)(66,173,131,158)(67,159,132,174)(68,175,133,160)(69,161,134,176)(70,177,135,162)(71,163,136,178)(72,179,137,164)(73,165,138,180)(74,181,139,166)(75,167,140,182)(76,183,113,168)(77,141,114,184)(78,185,115,142)(79,143,116,186)(80,187,117,144)(81,145,118,188)(82,189,119,146)(83,147,120,190)(84,191,121,148), (1,104)(2,36)(3,106)(4,38)(5,108)(6,40)(7,110)(8,42)(9,112)(10,44)(11,86)(12,46)(13,88)(14,48)(15,90)(16,50)(17,92)(18,52)(19,94)(20,54)(21,96)(22,56)(23,98)(24,30)(25,100)(26,32)(27,102)(28,34)(29,213)(31,215)(33,217)(35,219)(37,221)(39,223)(41,197)(43,199)(45,201)(47,203)(49,205)(51,207)(53,209)(55,211)(57,71)(58,164)(59,73)(60,166)(61,75)(62,168)(63,77)(64,142)(65,79)(66,144)(67,81)(68,146)(69,83)(70,148)(72,150)(74,152)(76,154)(78,156)(80,158)(82,160)(84,162)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111)(113,169)(114,128)(115,171)(116,130)(117,173)(118,132)(119,175)(120,134)(121,177)(122,136)(123,179)(124,138)(125,181)(126,140)(127,183)(129,185)(131,187)(133,189)(135,191)(137,193)(139,195)(141,170)(143,172)(145,174)(147,176)(149,178)(151,180)(153,182)(155,184)(157,186)(159,188)(161,190)(163,192)(165,194)(167,196)(198,212)(200,214)(202,216)(204,218)(206,220)(208,222)(210,224), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,115,15,129)(2,114,16,128)(3,113,17,127)(4,140,18,126)(5,139,19,125)(6,138,20,124)(7,137,21,123)(8,136,22,122)(9,135,23,121)(10,134,24,120)(11,133,25,119)(12,132,26,118)(13,131,27,117)(14,130,28,116)(29,70,43,84)(30,69,44,83)(31,68,45,82)(32,67,46,81)(33,66,47,80)(34,65,48,79)(35,64,49,78)(36,63,50,77)(37,62,51,76)(38,61,52,75)(39,60,53,74)(40,59,54,73)(41,58,55,72)(42,57,56,71)(85,147,99,161)(86,146,100,160)(87,145,101,159)(88,144,102,158)(89,143,103,157)(90,142,104,156)(91,141,105,155)(92,168,106,154)(93,167,107,153)(94,166,108,152)(95,165,109,151)(96,164,110,150)(97,163,111,149)(98,162,112,148)(169,221,183,207)(170,220,184,206)(171,219,185,205)(172,218,186,204)(173,217,187,203)(174,216,188,202)(175,215,189,201)(176,214,190,200)(177,213,191,199)(178,212,192,198)(179,211,193,197)(180,210,194,224)(181,209,195,223)(182,208,196,222) );

G=PermutationGroup([[(1,90,49,219),(2,220,50,91),(3,92,51,221),(4,222,52,93),(5,94,53,223),(6,224,54,95),(7,96,55,197),(8,198,56,97),(9,98,29,199),(10,200,30,99),(11,100,31,201),(12,202,32,101),(13,102,33,203),(14,204,34,103),(15,104,35,205),(16,206,36,105),(17,106,37,207),(18,208,38,107),(19,108,39,209),(20,210,40,109),(21,110,41,211),(22,212,42,111),(23,112,43,213),(24,214,44,85),(25,86,45,215),(26,216,46,87),(27,88,47,217),(28,218,48,89),(57,149,122,192),(58,193,123,150),(59,151,124,194),(60,195,125,152),(61,153,126,196),(62,169,127,154),(63,155,128,170),(64,171,129,156),(65,157,130,172),(66,173,131,158),(67,159,132,174),(68,175,133,160),(69,161,134,176),(70,177,135,162),(71,163,136,178),(72,179,137,164),(73,165,138,180),(74,181,139,166),(75,167,140,182),(76,183,113,168),(77,141,114,184),(78,185,115,142),(79,143,116,186),(80,187,117,144),(81,145,118,188),(82,189,119,146),(83,147,120,190),(84,191,121,148)], [(1,104),(2,36),(3,106),(4,38),(5,108),(6,40),(7,110),(8,42),(9,112),(10,44),(11,86),(12,46),(13,88),(14,48),(15,90),(16,50),(17,92),(18,52),(19,94),(20,54),(21,96),(22,56),(23,98),(24,30),(25,100),(26,32),(27,102),(28,34),(29,213),(31,215),(33,217),(35,219),(37,221),(39,223),(41,197),(43,199),(45,201),(47,203),(49,205),(51,207),(53,209),(55,211),(57,71),(58,164),(59,73),(60,166),(61,75),(62,168),(63,77),(64,142),(65,79),(66,144),(67,81),(68,146),(69,83),(70,148),(72,150),(74,152),(76,154),(78,156),(80,158),(82,160),(84,162),(85,99),(87,101),(89,103),(91,105),(93,107),(95,109),(97,111),(113,169),(114,128),(115,171),(116,130),(117,173),(118,132),(119,175),(120,134),(121,177),(122,136),(123,179),(124,138),(125,181),(126,140),(127,183),(129,185),(131,187),(133,189),(135,191),(137,193),(139,195),(141,170),(143,172),(145,174),(147,176),(149,178),(151,180),(153,182),(155,184),(157,186),(159,188),(161,190),(163,192),(165,194),(167,196),(198,212),(200,214),(202,216),(204,218),(206,220),(208,222),(210,224)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,115,15,129),(2,114,16,128),(3,113,17,127),(4,140,18,126),(5,139,19,125),(6,138,20,124),(7,137,21,123),(8,136,22,122),(9,135,23,121),(10,134,24,120),(11,133,25,119),(12,132,26,118),(13,131,27,117),(14,130,28,116),(29,70,43,84),(30,69,44,83),(31,68,45,82),(32,67,46,81),(33,66,47,80),(34,65,48,79),(35,64,49,78),(36,63,50,77),(37,62,51,76),(38,61,52,75),(39,60,53,74),(40,59,54,73),(41,58,55,72),(42,57,56,71),(85,147,99,161),(86,146,100,160),(87,145,101,159),(88,144,102,158),(89,143,103,157),(90,142,104,156),(91,141,105,155),(92,168,106,154),(93,167,107,153),(94,166,108,152),(95,165,109,151),(96,164,110,150),(97,163,111,149),(98,162,112,148),(169,221,183,207),(170,220,184,206),(171,219,185,205),(172,218,186,204),(173,217,187,203),(174,216,188,202),(175,215,189,201),(176,214,190,200),(177,213,191,199),(178,212,192,198),(179,211,193,197),(180,210,194,224),(181,209,195,223),(182,208,196,222)]])

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 2 2 8 14 14 28 28 28 56 2 2 2 4 4 28 28 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + - + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D7 SD16 C4○D4 D14 D14 D14 Dic14 C8⋊C22 D4⋊2D7 D4×D7 D8⋊D7 D7×SD16 kernel D4⋊Dic14 C4.Dic14 Dic7⋊C8 C8⋊Dic7 D4⋊Dic7 C7×D4⋊C4 C28⋊Q8 D4×Dic7 C2×Dic7 C7×D4 D4⋊C4 Dic7 C28 C4⋊C4 C2×C8 C2×D4 D4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 3 4 2 3 3 3 12 1 3 3 6 6

Matrix representation of D4⋊Dic14 in GL4(𝔽113) generated by

 112 91 0 0 72 1 0 0 0 0 1 0 0 0 0 1
,
 112 91 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 0 53 0 0 81 0 0 0 0 0 23 55 0 0 73 96
,
 112 91 0 0 72 1 0 0 0 0 7 60 0 0 18 106
G:=sub<GL(4,GF(113))| [112,72,0,0,91,1,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,91,1,0,0,0,0,112,0,0,0,0,112],[0,81,0,0,53,0,0,0,0,0,23,73,0,0,55,96],[112,72,0,0,91,1,0,0,0,0,7,18,0,0,60,106] >;

D4⋊Dic14 in GAP, Magma, Sage, TeX

D_4\rtimes {\rm Dic}_{14}
% in TeX

G:=Group("D4:Dic14");
// GroupNames label

G:=SmallGroup(448,295);
// by ID

G=gap.SmallGroup(448,295);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,254,219,226,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^28=1,d^2=c^14,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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